American Institute of Mathematical Sciences

Advanced
February  2016, 10(1): 45-52. doi: 10.3934/amc.2016.10.45

On extendability of additive code isometries

Serhii Dyshko 1,

1. 

IMATH, Université de Toulon, B.P. 20132, 83957 La Garde, France

Received  November 2014 Revised  July 2015 Published  March 2016

For linear codes, the MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a linear isometry of the whole space. But, in general, this is not the situation for nonlinear codes. In this paper codes over a vector space alphabet are considered. It is proved that if the length of such code is less than some threshold value, then an analogue of the MacWilliams Extension Theorem holds. One family of unextendable code isometries for the threshold value of code length is described.
Keywords: nonlinear codes., code isometries, Additive codes, subspaces covering, MacWilliams Extension Theorem.
Mathematics Subject Classification: Primary: 94B05; Secondary: 05B4.
Citation: Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
References:
[1]

S. V. Avgustinovich and F. I. Solov'eva, To the metrical rigidity of binary codes,, Probl. Inf. Transm., 39 (2003), 178.  doi: 10.1023/A:1025148221096.  Google Scholar

[2]

K. Bogart, D. Goldberg and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes,, Inf. Control, 37 (1978), 19.  doi: 10.1016/S0019-9958(78)90389-3.  Google Scholar

[3]

R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes,, J. Combin. Theory, 1 (1966), 96.  doi: 10.1016/S0021-9800(66)80007-8.  Google Scholar

[4]

I. Constantinescu and W. Heise, On the concept of code-isomorphy,, J. Geometry, 57 (1996), 63.  doi: 10.1007/BF01229251.  Google Scholar

[5]

H. Q. Dinh and S. R. López-Permouth, On the equivalence of codes over rings and modules,, Finite Fields Appl., 10 (2004), 615.  doi: 10.1016/j.ffa.2004.01.001.  Google Scholar

[6]

M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247.  doi: 10.1142/S0219498804000873.  Google Scholar

[7]

M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory Ser. A, 92 (2000), 17.  doi: 10.1006/jcta.1999.3033.  Google Scholar

[8]

J. Gruska, Quantum Computing,, McGraw-Hill, (1999).   Google Scholar

[9]

D. I. Kovalevskaya, On metric rigidity for some classes of codes,, Probl. Inf. Transm., 47 (2011), 15.  doi: 10.1134/S0032946011010029.  Google Scholar

[10]

J. Luh, On the representation of vector spaces as a finite union of subspaces,, Acta Math. Acad. Sci. Hungar., 23 (1972), 341.  doi: 10.1007/BF01896954.  Google Scholar

[11]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups,, Ph.D thesis, (1962).   Google Scholar

[12]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes,, North-Holland, (1977).   Google Scholar

[13]

F. Solov'eva, T. Honold, S. Avgustinovich and W. Heise, On the extendability of code isometries,, J. Geometry, 61 (1998), 2.  doi: 10.1007/BF01237489.  Google Scholar

[14]

H. N. Ward and J. A. Wood, Characters and the equivalence of codes,, J. Combin. Theory Ser. A, 73 (1996), 348.  doi: 10.1016/S0097-3165(96)80011-2.  Google Scholar

[15]

J. A. Wood, Duality for modules over finite rings and applications to coding theory,, Amer. J. Math., 121 (1999), 555.  doi: 10.1353/ajm.1999.0024.  Google Scholar

[16]

J. A. Wood, Foundations of linear codes defined over finite modules: The extension theorem and the MacWilliams identities,, in Codes over Rings (ed. P. Sóle), (2009), 124.  doi: 10.1142/9789812837691_0004.  Google Scholar

show all references

References:
[1]

S. V. Avgustinovich and F. I. Solov'eva, To the metrical rigidity of binary codes,, Probl. Inf. Transm., 39 (2003), 178.  doi: 10.1023/A:1025148221096.  Google Scholar

[2]

K. Bogart, D. Goldberg and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes,, Inf. Control, 37 (1978), 19.  doi: 10.1016/S0019-9958(78)90389-3.  Google Scholar

[3]

R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes,, J. Combin. Theory, 1 (1966), 96.  doi: 10.1016/S0021-9800(66)80007-8.  Google Scholar

[4]

I. Constantinescu and W. Heise, On the concept of code-isomorphy,, J. Geometry, 57 (1996), 63.  doi: 10.1007/BF01229251.  Google Scholar

[5]

H. Q. Dinh and S. R. López-Permouth, On the equivalence of codes over rings and modules,, Finite Fields Appl., 10 (2004), 615.  doi: 10.1016/j.ffa.2004.01.001.  Google Scholar

[6]

M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes,, J. Algebra Appl., 3 (2004), 247.  doi: 10.1142/S0219498804000873.  Google Scholar

[7]

M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem,, J. Combin. Theory Ser. A, 92 (2000), 17.  doi: 10.1006/jcta.1999.3033.  Google Scholar

[8]

J. Gruska, Quantum Computing,, McGraw-Hill, (1999).   Google Scholar

[9]

D. I. Kovalevskaya, On metric rigidity for some classes of codes,, Probl. Inf. Transm., 47 (2011), 15.  doi: 10.1134/S0032946011010029.  Google Scholar

[10]

J. Luh, On the representation of vector spaces as a finite union of subspaces,, Acta Math. Acad. Sci. Hungar., 23 (1972), 341.  doi: 10.1007/BF01896954.  Google Scholar

[11]

F. J. MacWilliams, Combinatorial Properties of Elementary Abelian Groups,, Ph.D thesis, (1962).   Google Scholar

[12]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes,, North-Holland, (1977).   Google Scholar

[13]

F. Solov'eva, T. Honold, S. Avgustinovich and W. Heise, On the extendability of code isometries,, J. Geometry, 61 (1998), 2.  doi: 10.1007/BF01237489.  Google Scholar

[14]

H. N. Ward and J. A. Wood, Characters and the equivalence of codes,, J. Combin. Theory Ser. A, 73 (1996), 348.  doi: 10.1016/S0097-3165(96)80011-2.  Google Scholar

[15]

J. A. Wood, Duality for modules over finite rings and applications to coding theory,, Amer. J. Math., 121 (1999), 555.  doi: 10.1353/ajm.1999.0024.  Google Scholar

[16]

J. A. Wood, Foundations of linear codes defined over finite modules: The extension theorem and the MacWilliams identities,, in Codes over Rings (ed. P. Sóle), (2009), 124.  doi: 10.1142/9789812837691_0004.  Google Scholar

[1]

Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124
[2]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118
[3]

Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020129
[4]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044
[5]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038
[6]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395
[7]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045
[8]

Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, 2021, 15 (2) : 207-218. doi: 10.3934/amc.2020053
[9]

Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065
[10]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055
[11]

Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054
[12]

Shanding Xu, Longjiang Qu, Xiwang Cao. Three classes of partitioned difference families and their optimal constant composition codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020120
[13]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049
[14]

Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng. A construction of $ p $-ary linear codes with two or three weights. Advances in Mathematics of Communications, 2021, 15 (1) : 9-22. doi: 10.3934/amc.2020039
[15]

Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $ 6lp^s $. Advances in Mathematics of Communications, 2021, 15 (1) : 167-189. doi: 10.3934/amc.2020051
[16]

Saadoun Mahmoudi, Karim Samei. Codes over $ \frak m $-adic completion rings. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020122
[17]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374
[18]

Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127
[19]

Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041
[20]

Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences